A Core Course on Modeling

1. define Right concepts? Right problem? conceptualize Right model? formalize Right outcome? execute Right answer? conclude A Core Course on Modeling Week 1- No Model Without a Purpose   The modeling process  1 formulate purpose identify entities choose relations obtain values formalize relations operate model obtain result present result interpret result

2. A Core Course on Modeling Week 4-The Function of Functions    Contents     • What is a Formal Model? • A Practical Route to Formal Models • Example 1: The Detergent Problem • Example 2: The Chimney Sweepers Problem • Example 3: The Peanut Butter Problem • The relation wizard • The function selector • Summary • References to lecture notes + book • References to quiz-questions and homework assignments (lecture notes)

3. A Core Course on Modeling Week 4-The Function of Functions    What is a Formal Model?     3 What is the meaning of + ? the intuition of ‘addition’, ‘accumulation’: resistors: Rtot = 1/(1/R1 + 1/R2) springs: Ctot = C1 + C2 resistors: Rtot = R1 + R2 springs: Ctot = 1/(1/C1+1/C2)

4. A Core Course on Modeling Week 4-The Function of Functions    A Practical Route to Formal Models     4 Heuristics to arrive at formal expressions: • meaningful names • chain of dependencies • todo-list • dimensional analysis • wisdom of the crowds

5. A Core Course on Modeling Week 4-The Function of Functions     The Detergent Problem     5 “What is the total amount of detergent annually dumped in the Environment in the Netherlnds?

6. A Core Course on Modeling Week 4-The Function of Functions     The Detergent Problem     6 relations dimensions assumptions todo amAnDetDmp = f(nrAnWshs , detPWsh) [kg / year] = F([wash / year] , [kg / wash]) amAnDetDmp nrAnWshs detPWsh Det:detergent; An: annual; Wsh: wash; Fam: family; P: people; am: amount; Dmp: dump

7. A Core Course on Modeling Week 4-The Function of Functions     The Detergent Problem     7 relations dimensions assumptions todo amAnDetDmp = nrAnWshs * detPWsh [kg / year] = [wash / year] * [kg / wash] amAnDetDmp nrAnWshs detPWsh Det:detergent; An: annual; Wsh: wash; Fam: family; P: people; am: amount; Dmp: dump

8. A Core Course on Modeling Week 4-The Function of Functions     The Detergent Problem     8 relations dimensions assumptions todo amAnDetDmp = nrAnWshs * detPWsh [kg / year] = [wash / year] * [kg / wash] Washing laundry is the only way detergent gets into the environment amAnDetDmp nrAnWshs detPWsh nrAnWshsPFam nrFamIH nrPIH nrPPFam No institutional laundry washing, only families nrAnWshs = nrAnWshsPFam * nrFamIH [wash / year] = [wash / (fam *year)] * [fam] Everybody belongs to exactly one family: families are disjoint nrFamIH = nrPIH / nrPPFam [fam] = [people] / [people / fam] todo list is empty  model is ready common knowledge nrPIH = 17  0.5 million [people] public domain nrPPFam = 2.2  0.2 [people/fam] nrAnWshsPFam = ????? [wash / year] wisdom of the crowds detPWsh = 0.17  0.03 [kg / wash] wisdom of the crowds Det:detergent; An: annual; Wsh: wash; Fam: family; P: people; am: amount; Dmp: dump

9. A Core Course on Modeling Week 4-The Function of Functions     The Detergent Problem     9 todo * amAnDetDmp amAnDetDmp nrAnWshs detPWsh nrAnWshsPFam nrFamIH nrPIH nrPPFam detPWsh nrAnWshs * / nrFamIH nrAnWshsPFam nrPPFam nrPIH

10. A Core Course on Modeling Week 4-The Function of Functions     The Detergent Problem     10 todo * amAnDetDmp 1.31… million amAnDetDmp nrAnWshs detPWsh nrAnWshsPFam nrFamIH nrPIH nrPPFam detPWsh nrAnWshs * 0.170.03 772…million / 7.72 … million nrFamIH nrAnWshsPFam 10020 nrPPFam nrPIH 2.20.2 170.5 million

11. A Core Course on Modeling Week 4-The Function of Functions    The Chimney Sweepers Problem     11 “How many chimney sweepers work in Amsterdam?”

12. A Core Course on Modeling Week 4-The Function of Functions    The Chimney Sweepers Problem     12 relations dimensions assumptions todo nrChSwIA = nrChIA * nrSwPCh [Sw / A] = [Ch / A] * [SW / Ch] Amsterdam ch.-sweepers sweep Amsterdam chimneys only nrChSwIA nrChIA nrSwPCh nrChPFam nrFamIA nrPIA nrPPFam ch.-sweepers sweep only chimneys on family houses nrChIA = nrChPFam * nrFamIA [Ch / A] = [Ch / Fam] * [Fam / A] nrPPFam is the same everywhere (does not depend on ‘Amsterdam’) nrFamIA = nrPIA / nrPPFam [Fam / A] = [P / A] / [P / Fam] common knowledge nrPIA = 790000 [P] public domain nrPPFam = 2.2  0.2 [P/Fam] nrChPFam (= 1/nrFamPCh) =0.10.02 [Ch/Fam] wisdom of the crowds Sw=sweeper; Ch=chimney;A=Amsterdam;Fam=Family;P=people;Se=Service;

13. A Core Course on Modeling Week 4-The Function of Functions    The Chimney Sweepers Problem     13 relations dimensions assumptions todo nrSwPCh = nrSwPSe * nrSePCh [Sw / Ch] = [Sw*year/Se] * [Se/(Ch*year)] Introduce time to associate sweeper’s ca-pacity to chimney’s need nrChSwIA nrChIA nrSwPCh nrChPFam nrFamIA nrPIA nrPPFam nrSwPSe nrSePCh timeP1Se timeP1Sw assume average times (i.e., no season influences etc.) nrSwPSe = timeP1Se / timeP1Sw [Sw * year / Se] = [hour / Se] / [hour / (Sw*year)] wisdom of the crowds timeP1Se = 20.25 hour / Se work year = 1600 hours timeP1Sw = 1200100 hour / Sw * year) insurance requirement nrSePCh = 1 Se /( Ch * year) todo list is empty  model is ready but what does it mean ?  NOTHING, since we formulated no purpose Sw=sweeper; Ch=chimney;A=Amsterdam;Fam=Family;P=people;Se=Service;

14. A Core Course on Modeling Week 4-The Function of Functions    The Chimney Sweepers Problem     14 What purposes could we think of: begin a professional journal … so: we only need to know if NrChSwIA > 300 ChSw convention meeting in the Restaurant ‘the Swinging Sweeper?’ so: we only need to know if NrChSwIA <50 form efficient ‘Chimney and Sewage Control and Service Units’? so: we only need to know if NrChSwIA is between 50 and 60…

15. A Core Course on Modeling Week 4-The Function of Functions    The Chimney Sweepers Problem     15 To assess credibility of a model: • actual measurements ? • second, independent model ? Such as … • how much soot and ashes are disposed of by the municipal Ash & Soot Depot? • how often do you see a chimney sweeper at work? • how much money do people in A. spend annually in cleaning their chimneys? check latter case:

16. PB A Core Course on Modeling Week 4-The Function of Functions     The Peanut Butter Problem     16 “How to get rich by selling a new brand of peanut butter?”

17. A Core Course on Modeling Week 4-The Function of Functions     The Peanut Butter Problem     17 relations dimensions assumptions todo profit income expenses pricePerItem nrSoldItems nrSoldTotal marketShare profit = income - expenses [Euro / year] = [Euro/year] no taxes, no inflation income = pricePerItem * nrSoldItems [Euro / year] = [Euro/myPB] * [myPB/year] no discount with larger quantities per purchase nrSoldItems = nrSoldTotal * marketShare [myPB / year] = [allPB / year] * [myPB/allPB] my PB will not increase the total market from a neutral marketing bureau nrSoldTotal = … [allPB/year] pricePerItem = … marketShare = …

18. 1 marketShare 0 pricePerItem A Core Course on Modeling Week 4-The Function of Functions     The Peanut Butter Problem     18 Approach 1: glass box ?

19. 1 marketShare 0 pricePerItem A Core Course on Modeling Week 4-The Function of Functions     The Peanut Butter Problem     19 Approach 1: glass box What mechanism determines marketShare(pricePerItem)? • monotonically decreasing • between 0 and 1 • asymptote: marketShare0 if pricePerItem  • asymptote: marketShare1 if pricePerItem -  • what sort of mathematical dependency ???

20. cheapest competitor most expensive competitor 1 marketShare 0 pricePerItem A Core Course on Modeling Week 4-The Function of Functions     The Peanut Butter Problem     20 Approach 1: glass box

21. cheapest competitor most expensive competitor 1 marketShare 0 pricePerItem A Core Course on Modeling Week 4-The Function of Functions     The Peanut Butter Problem     21 Approach 1: glass box

22. cheapest competitor most expensive competitor 1 marketShare 0 pricePerItem A Core Course on Modeling Week 4-The Function of Functions     The Peanut Butter Problem     22 Approach 1: glass box

23. 1 marketShare 0 pricePerItem A Core Course on Modeling Week 4-The Function of Functions     The Peanut Butter Problem     23 Approach 2: black box

24. 1 marketShare 0 pricePerItem A Core Course on Modeling Week 4-The Function of Functions     The Peanut Butter Problem     24 Approach 2: black box

25. A Core Course on Modeling Week 4-The Function of Functions     The Peanut Butter Problem     25 Conceptual model  formal model: while (purpose not satisfied): • identify quantity q needed for purpose • q  to-do list • while todo (not empty): • take a quantity r from todo list • think: what does r depend on? • if r independent  substitute constant value with uncertainty bounds • else give expression for r • if possible, use dimensional analysis • propose mathematical expression r = f(s1, s2, s3, …) • think about assumptions • verify dimensions • add newly introduced quantities s1, s2, s3, … to the todo list • todo list is empty: evaluate model • check if purpose is satisfied; if not, refine model • ready

26. A Core Course on Modeling Week 4-The Function of Functions     The Peanut Butter Problem     26 To get help in the translation to mathematical relations: relation wizard function selector

27. cheapest competitor most expensive competitor 1 marketShare 0 pricePerItem A Core Course on Modeling Week 4-The Function of Functions     The Peanut Butter Problem     27 Approach 1: glass box

28. A Core Course on Modeling Week 4-The Function of Functions     Summary     28 • Conceptual model formal model : not in a formally provable correct way; • Appropriate naming • Structure • Chain of dependencies: the formal model as a directed acyclic graph; • What mechanism? • What quantitiesdrive this mechanism? • What is the qualitative behavior of the mechanism? • What is the mathematical expression to describe this mechanism? • To-do-list: all intermediate quantities are found and elaborated in turn; • Formationofmathematical expressions: • dimensional analysis mathematical expressions, e.g in the case of proportionality • the Relation Wizardcan help finding appropriate fragments of mathematics; • the Function Selectorcan help finding an appropriate expression for a desired behavior; • wisdom of the crowdscan help improve the accuracy of guessed values;

29. A Core Course on Modeling Week 4-The Function of Functions     The Peanut Butter Problem     29 Peanut Butter example: income = pricePerItem * nrSoldItems nrSoldItems = nrSoldTotal * marketShare marketShare = f(pricePerItem)

30. A Core Course on Modeling Week 4-The Function of Functions     The Peanut Butter Problem     30 Revisit the peanut butter example: inc = pPI * nSI (inc=income; pPI=pricePerItem; nSI=nrSoldItems) nSI = nST * mSh (nST=nrSoldTotal; mSh=marketShare) mSh = f(pPI) So: inc = pPI * nST * mSh(pPI) Realized that nST = i nST(i), i ranges over customers. So: inc = pPI * mSh(pPI) * i nST(i) Improved model: inc = pPI * i nST(i) * mSh(pPI , i)